While such \(\varepsilon \) -nets can be constructed by taking any maximal set of points with disjoint \(\varepsilon /2\) -balls,
the geometric generation process of such a maximal set is based on the close connection between the growth rate of volumes of balls in manifolds and Ricci curvature.
Furthermore, the connection between volume and Ricci curvature can be extended to more general measures and to
the generalized Ricci curvature developed by Lott-Villani [1]} and Sturm [2]}. Therefore, it is natural to seek and generalize the results of Grove and Petersen to metric measure spaces. Such an extension of the classical case construction to the metric measure spaces context does, indeed exists [3]}.
